De Morgan's theorems: Foundations — Hardware deep dive

Before you can read the parent note, you must own each symbol it throws at you. This page builds them one at a time, from absolutely nothing, in the order they depend on each other.

1. The value: a Boolean is only ever 0 or 1

The picture: think of a single light switch. It is either OFF ( $0$ ) or ON ( $1$ ) — never "half". There is no $0.5$ , no $2$ , no negatives. That is the entire number world we live in for this topic.

Why the topic needs it: every truth table in the parent note has cells that are only $0$ or $1$ . Every proof works by checking "does the left side give the same $0$ / $1$ as the right side, on every row?". If you don't accept "only two values", none of the tables make sense.

2. The variable: a named box holding a Boolean

The picture: a box with a nameplate " $A$ " on it. Right now it's empty; when we "evaluate", we drop a $0$ or a $1$ inside.

Why the topic needs it: the theorems must hold for any inputs. Writing $\overline{A \cdot B} = \overline{A} + \overline{B}$ with letters means "no matter which $0$ / $1$ you put in $A$ and $B$ , this is true." Letters let one line of algebra cover all four combinations at once.

Recall How many different value-combinations do two variables

$A, B$ have? Four: $(0, 0), (0, 1), (1, 0), (1, 1)$ . That is why every 2-variable truth table has exactly 4 rows. In general $n$ variables give $2^{n}$ rows.

3. NOT — the bar that flips a value

The picture: a see-saw. Push the $0$ end down and the $1$ end comes up. NOT is that flip — one input, one output, always the reverse.

Why the scope (bar length) matters — read this twice: The bar is not just decoration; its length tells you what it applies to.

$\overline{A} \cdot \overline{B}$ — two short bars — means "flip $A$ , flip $B$ , then AND them."
$\overline{A \cdot B}$ — one long bar — means "AND them first, then flip the single result."

These are different operations and (usually) give different answers — that difference is the entire point of De Morgan.

4. AND — the "both" operation ( $\cdot$ )

Why a dot? Because AND behaves exactly like multiplying $0$ s and $1$ s: $0 \cdot 0 = 0, 0 \cdot 1 = 0, 1 \cdot 1 = 1$ . Choosing the multiplication symbol lets you reuse arithmetic instincts.

Its full behaviour (its truth table):

$A$	$B$	$A \cdot B$
0	0	0
0	1	0
1	0	0
1	1	1

The picture: two switches wired in series (one after the other) in a torch circuit. The bulb lights only if both switches are closed. Miss one, and the current is blocked.

Why the topic needs it: Theorem 1 is entirely about "what is the NOT of an AND?".

5. OR — the "either" operation ( $+$ )

Why a plus? Because for the three "normal" cases it looks like addition: $0 + 0 = 0, 0 + 1 = 1$ . The one oddity is $1 + 1$ , which is $1$ (not $2$ — remember, values can only be $0$ or $1$ ). So it's "addition that saturates at 1".

$A$	$B$	$A + B$
0	0	0
0	1	1
1	0	1
1	1	1

The picture: two switches wired in parallel (side by side). Closing either one completes a path and lights the bulb.

Why the topic needs it: Theorem 2 is "what is the NOT of an OR?", and the answer of Theorem 1 is an OR. These two operators are the pair that swap places.

6. Equality of expressions — what " $=$ " means here

The picture: two vending machines. Press the same buttons on both; if they always drop the same snack for every button-combination, the machines are equivalent — even if their insides are wired differently. That is exactly how the parent note proves each theorem: build both sides' columns and check they agree on all $4$ rows.

Why the topic needs it: this is the only proof method used. " $=$ " in Boolean algebra is a claim to be tested against a table, not something to accept on sight.

7. Putting it together — reading the theorems

Now every symbol in the parent note is earned. Read the headline formula out loud in plain words:

$NOT of (A AND B) \overline{A A \cdot B A} = not A \overline{A} OR + not B \overline{B}$

"The NOT of ( $A$ AND $B$ ) equals (not $A$ ) OR (not $B$ )." Because you now know the bar wraps the whole AND, and that flipping an AND turns it into an OR, this sentence is readable — and checkable with the equality rule from section 6.

Prerequisite map

Equipment checklist

Cover the right side and test whether you're ready for the parent note.

A Boolean value can be exactly which values

Only

0

1

— nothing else

What does the bar

\overline{A}

Flips the value:

\overline{0} = 1

\overline{1} = 0

\overline{\overline{A}}

equals

A

(double negation returns the original)

A \cdot B

(AND) is

1

only when

Both

A

and

B

are

1

A + B

(OR) is

0

only when

Both

A

and

B

are

0

Why is

1 + 1 = 1

in Boolean OR

Values saturate at

1

; there is no

2

Difference between

\overline{A \cdot B}

and

\overline{A} \cdot \overline{B}

Bar length/scope — long bar negates the whole AND result; short bars negate each variable first

Two Boolean expressions are equal when

They give the same output for every input row of the truth table

How many rows does a 2-variable truth table have

Four (

2^{2}

)

Series switches model which gate

AND (all needed)

Parallel switches model which gate

OR (either one)

Connections

De Morgan's theorems — the parent topic these foundations unlock.
Logic Gates - AND OR NOT — the physical gates for the three operations built here.
Boolean Algebra Laws — where equality and these operators are used as a full algebra.
Complement of a Function — extends NOT-over-a-group to whole functions.
NAND and NOR Universal Gates — built from NOT + AND/OR.
Bubble Pushing — the graphical face of bar-scope and negation.
Karnaugh Maps — another way to reason about the same truth tables.